Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $n = \dfrac{x - 9}{x - 1} \times \dfrac{-2x^2 - 18x + 20}{x^2 - 11x + 18} $
Explanation: First factor out any common factors. $n = \dfrac{x - 9}{x - 1} \times \dfrac{-2(x^2 + 9x - 10)}{x^2 - 11x + 18} $ Then factor the quadratic expressions. $n = \dfrac {x - 9} {x - 1} \times \dfrac {-2(x - 1)(x + 10)} {(x - 9)(x - 2)} $ Then multiply the two numerators and multiply the two denominators. $n = \dfrac {(x - 9) \times -2(x - 1)(x + 10) } {(x - 1) \times (x - 9)(x - 2) } $ $n = \dfrac {-2(x - 1)(x + 10)(x - 9)} {(x - 9)(x - 2)(x - 1)} $ Notice that $(x - 9)$ and $(x - 1)$ appear in both the numerator and denominator so we can cancel them. $n = \dfrac {-2(x - 1)(x + 10)\cancel{(x - 9)}} {\cancel{(x - 9)}(x - 2)(x - 1)} $ We are dividing by $x - 9$ , so $x - 9 \neq 0$ Therefore, $x \neq 9$ $n = \dfrac {-2\cancel{(x - 1)}(x + 10)\cancel{(x - 9)}} {\cancel{(x - 9)}(x - 2)\cancel{(x - 1)}} $ We are dividing by $x - 1$ , so $x - 1 \neq 0$ Therefore, $x \neq 1$ $n = \dfrac {-2(x + 10)} {x - 2} $ $ n = \dfrac{-2(x + 10)}{x - 2}; x \neq 9; x \neq 1 $